Math 440, Fall 2014, Assignment 4

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

Carefully define the following terms, then give one example and one non-example of each:

1. Subspace topology (on a subset $A$ of a topological space $X$).
2. Function continuous at a given point.
3. Continuous function.
4. Homeomorphism (from one space top another).
5. Homeomorphic (spaces).
6. Topological property.

Carefully state the following theorems (you do not need to prove them):

1. Theorem characterizing continuous functions (Theorem 7.2).
2. Theorem on composition of continuous functions.
3. Theorem on restriction of continuous functions (Theorem 7.5).
4. Theorem concerning continuity of functions defined on unions (Theorem 7.6).
5. Theorem concerning restriction of codomains (Theorem 7.7).

Solve the following problems:

1. Problems 6C, 7A, 7C, and 7G.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Subspace Topology:

   Let \((X,\tau)\) be a topological space, and let \(A \subseteq X\). The subspace topology of \(X\) on \(A\) is the topological space \((A,\tau_A)\), where \(\tau_A = \{A \cap t| t\in \tau\}\).

   Example:

   Non-Example:

2. Function Continuous at a Given Point:

   Let \(f:X \rightarrow Y\) be a function between topological spaces. \(f\) is continuous at \(x \in X\) if for every \(V \subseteq Y\) that contains \(f(x)\) (i.e. \(f(x) \in V\), then \(f^{-1}(V)\) is open in \(X\).

   Example:

   Non-Example:

3. Continuous Function:

   Let \(X\) and \(Y\) be topological spaces. A function \(f : X \rightarrow Y\) is continuous if it is continuous at every point of \(X\).

   Example:

   Non-Example:

4. Homeomorphism:

   A homeomorphism from \(X\) to \(Y\) is a continuous bijective function with a continuous inverse.

   Example:

   Non-Example:

5. Homeomorphic Spaces:

   Two topological spaces \(X\) and \(Y\)

   Example:

   Non-Example:

6. Topological Property:

   A topological property is a property that is preserved on homeomorphism. That is, if \((X,\tau)\) and \((Y,\sigma)\) are homeomorphic spaces, then if \(X\) has a property, \(Y\) has the same property.

   Example:

   Cardinality.

   Non-Example:

Theorems:

1. Theorem Characterizing Continuous Functions

   Let \(f: X \rightarrow Y\). The following conditions are equivalent:
   \(f\) is continuous
   For each open set \(H\) in \(Y\), \(f^{-1}(H)\) is open in \(X\).
   For each closed set \(K\) in \(Y\), \(f^{-1}(K)\) is closed in \(X\).
   For each \(E \subset X\), \(f(Cl_X(E) \subset Cl_Y f(E)\).

2. Theorem on Composition of Continuous Functions

   If \(f:X\rightarrow Y\) is continuous, and \(g: Y\rightarrow Z\) is continuous, then \(g \circ f : X \rightarrow Z\) is continuous.

3. Theorem on Restriction of Continuous Functions

   If \(f: X \rightarrow Y\) is continuous, and \(A \subseteq X\), then \(f | A\) is continuous.

4. Theorem Concerning Continuity of Functions Defined on Unions

   If \(X = A \cup B\), and \(A\) and \(B\) are both open or both closed, and \(f|A\) and \(f|B\) are both continuous, then \(f\) is continuous.

5. Theorem Concerning Restriction of Codomains

   Let \(f: X \rightarrow Y\), and \(Y \subset Z\). Then \(f\) from \(X\) to \(Y\) if and only if \(f\) is continuous as a map from \(X\) to \(Z\).